Modelling and Filtering for Non-Markovian Quantum Systems

Modelling and Filtering for Non-Markovian Quantum Systems ?

arXiv:1704.00986v1 [quant-ph] 4 Apr 2017

Shibei Xue a , Thien Nguyen b , Matthew R. James b , Alireza Shabani c , Valery Ugrinovskii a , Ian R. Petersen b a

School of Information Technology and Electrical Engineering, University of New South Wales at the Australian Defence Force Academy, Canberra, ACT 2600, Australia b

Research School of Engineering, Australian National University, Canberra, ACT 2600, Australia c

Quantum Artificial Intelligence Lab, Google, 340 Main St. Venice, CA 90291, U.S.A.

Abstract This paper presents an augmented Markovian system model for non-Markovian quantum systems. In this augmented system model, ancillary systems are introduced to play the role of internal modes of the non-Markovian environment converting white noise to colored noise. Consequently, non-Markovian dynamics are represented as resulting from direct interaction of the principal system with the ancillary system. To demonstrate the utility of the proposed augmented system model, it is applied to design whitening quantum filters for non-Markovian quantum systems. Examples are presented to illustrate how whitening quantum filters can be utilized for estimating non-Markovian linear quantum systems and qubit systems. In particular, we showed that the augmented Markovian formulation can be used to theoretically model the environment for an observed nonMarkovian behavior in a recent experiment on quantum dots [10].

Key words: Non-Markovian quantum systems; Quantum stochastic differential equation; Whitening quantum filter; Quantum Kalman filter; Quantum colored noise.

1

Introduction

Control of open quantum systems has been rapidly advancing quantum information technology in recent years [5,25,33,18,29,1,8,28], where an open quantum system refers to a quantum system interacting with an environment or other quantum systems. ? This paper was not presented at any IFAC meeting. This research was supported under Australian Research Councils Discovery Projects and Laureate Fellowships funding schemes (Projects DP140101779 and FL110100020), the Chinese Academy of Sciences President’s International Fellowship Initiative (No. 2015DT006), and the Air Force Office of Scientific Research (AFOSR) under agreement FA2386-161-4065. Corresponding author S. Xue, Tel. : (+61) 02-62689461 Email addresses: [email protected] (Shibei Xue), [email protected] (Thien Nguyen), [email protected] (Matthew R. James), [email protected] (Alireza Shabani), [email protected] (Valery Ugrinovskii), [email protected] (Ian R. Petersen).

Preprint submitted to Automatica

Among open quantum systems, the most widely investigated class of quantum systems is that comprising quantum systems coupled with memoryless environment. Evolution of such systems can be described by master equations [5] in the Schr¨odinger picture and Langevin equaitons [5] or quantum stochastic differential equations [4] in the Heisenberg picture. Both mathematical models give rise to Markovian dynamics, thus this class of open quantum systems is commonly referred to as Markovian quantum systems. In addition, Markovian quantum systems can be coupled to a field satisfying a singular commutation relation, e.g., quantum white noise [12]. In terms of its stochastic description, the field has an independent increment over an infinitesimal time interval, which satisfies a non-demolition condition and can be measured to continuously extract information of the quantum system [3]. These properties of the environment are analogous to properties of a classical white noise and have served as a foundation for the well established quantum filtering theory aimed at estimation of Markovian quantum systems [4]. Such a theory underpins a number of successful control applications such

5 April 2017

is defined on an augmented Hilbert space. Also, we introduce a spectral factorization method to determine the structure of linear ancillary systems to ensure that its fictitious output has a power spectral density which is identical to that of the non-Markovian environment under consideration. Nevertheless, while these elements of our model follow the classical system modelling, the proposed model has a distinctively quantum feature in that the quantum plant and its non-Markovian environment mutually influence each other. This feature distinguishes quantum system-environment interactions from the classical case where the classical colored noise disturbs a plant but not vice versa [20]. To account for this special feature of non-Markovian quantum systems, in the proposed model the ancillary system is coupled to the principal system via their direct interactions rather than the field-mediated connection; cf. [14].

the design of real-time feedback control laws for cooling a quantum particle [7], and stabilizing states [24] or entanglement [43,47] of a quantum system. However, many problems of interest involve more complicated environmental influences, which cannot be handled within the Markovian setting and require treating the environment as quantum colored noise. This necessitates the investigation of non-Markovian behavior of quantum systems [32,40,6,39,2,38,37]. A non-Markovian quantum system is a quantum system interacting with an environment with memory effects. To describe dynamics of non-Markovian quantum systems involving quantum colored noise, several models have been developed, for example, non-Markovian Langevin equations where non-Markovian effects are embedded in a memory kernel function [32] and time-convolutionless master equations where a time-varying damping function characterizes the non-Markovian damping processes [5], etc. However, these existing models for non-Markovian quantum systems are not compatible with the quantum filtering theory. In addition, unlike the quantum white nose, the quantum colored noise does not satisfy the singular commutation relations. For that reason, non-Markovian models are difficult to use for processing quantum measurements. Once the quantum colored noise is measured, the states of the quantum system interacting with this noise will be demolished.

To describe the augmented Markovian system model for the non-Markovian quantum system, the paper adopts so-called (S, L, H) description, where the internal energy, the couplings to the environment and the scattering process of the environmental field for a quantum system are captured by a Hamiltonian H, a coupling operator L, and a scattering matrix S, respectively. An advantage of this approach is that it allows to describe system-environment interactions systematically using the formalism of quantum stochastic differential equations. To demonstrate this advantage and the utility of the proposed augmented Markovian modelling of non-Markovian quantum dynamics, we show how the proposed approach can be utilized to obtain whitening quantum filters for non-Markovian linear quantum systems and qubit systems. As an example application, this augmented Markovian model is utilized to explore quantum colored noise in a recent experiment for a hybrid solid-state quantum system [10].

A standard approach in classical control systems analysis and design is whitening of the colored noise by introducing additional dynamics so as to express the system with non-Markovian effects of colored noise as an augmented system model governed by a white noise. Thus a filter for the system involving colored noise can be constructed [19]; such filter is often referred to as whitening filter. Similar ideas have been explored for quantum systems as well. A pseudo-mode method was proposed for effectively simulating non-Markovian effects by using a Monte Carlo wave-function [15,23], which was applied to model the energy transfer process in photosynthetic complexes [30]. Also, the dynamics of nonMarkovian quantum systems can be described by using a hierarchy equation approach [21], which has been applied to indirect measurement of a non-Markovian quantum system [31]. An augmented system approach has been applied to obtain a quantum filter for quantum systems interacting with non-classical fields using a fieldmediated connection method in a situation where the non-Markovian system does not introduce backaction on the environment [14].

The paper is organized as follows. In section 2, we briefly review a general description of Markovian quantum systems. Based on this description, an augmented Markovian system model is presented for non-Markovian quantum systems in section 3, where a spectral factorization method is proposed to obtain an ancillary linear quantum system model for a quantum environment with a given spectrum. Next, the application to derivation of a whitening quantum filter is presented in section 4 where examples of filters for linear non-Markovian quantum systems and single qubit systems are obtained. In section 5, an augmented Markovian system model is utilized to obtain an improved model for an experiment involving a hybrid solid-state system. Finally, conclusions and discussions are given in Section 6.

In this paper we present a systematic augmented Markovian system approach to modelling non-Markovian quantum systems. To capture effects of the nonMarkovian environment, we introduce ancillary systems to augment a principal system of interest, which are realized by linear open quantum systems. Compared to the principal system, the augmented system model

2

2

Optical Cavity

A brief review of the Markovian quantum system model

Output field

Probing Field

In this section, we will briefly introduce some standard facts about Markovian quantum systems. For more details, we refer the reader to references [11,4]. 2.1

Fig. 1. A standard Markovian quantum system model, i.e., optical modes trapped in a cavity probed by a white noise field.

Quantum white noise

In quantum physics, it is customary to describe an environment field as the Fourier transform of the annihilation operator of the field acting on the so-called Fock space F [11]; that is 1 b(t) = √ 2π

Z

Markovian quantum systems can be described by quantum stochastic differential equations. Consider a Markovian quantum system and the associated triple (S, L, H) characterizing its scattering matrix, the coupling operator and the Hamiltonian, respectively. In quantum mechanics, both the coupling operator and the Hamiltonian are represented as operators mapping an underlying Hilbert space h into itself, As for the scattering matrix S, from now on, we will assume S = I since for simplicity the scattering process will not be considered here.

+∞

b(ω)e−iωt dω,

(1)

−∞

The operator on the Fock space defined by (1) is called the quantum white noise field operator. It satisfies singular commutation relations [b(t), b† (t0 )] = δ(t − t0 ),

[b(t), b(t0 )] = 0,

The unitary evolution operator Θ(t) of this Markovian quantum system is defined on the tensor product Hilbert space h ⊗ F, it is known to satisfy the following quantum It¯o stochastic differential equation

(2)

where [·, ·] is the commutator of operators, i.e., [f, g] = f g − gf for two operators f and g with suitable domain and image spaces. The symbol † denotes the complex conjugate of an operator.


1 − iH + L† L dt + dB † (t)L − L† dB(t) Θ(t). 2 (4) In the Heisenberg picture, all quantum mechanical quantities of interest affected by the system evolution are operators that initially defined on the Hilbert space h, and evolve on the Hilbert space h ⊗ F. Specifically, an arbitrary operator X defined on h gives rise to the evolution jt (X) of operators on the Hilbert space h ⊗ F defined as jt (X) = Θ† (t)XΘ(t). From (4), it follows that this evolution satisfies the quantum stochastic differential equation

dΘ(t) =

The quantum white noise field can be interpreted as a quantum stochastic process. Associated R t with the field b(t) are an integrated operator B(t) = t0 b(τ )dτ , known as B(t) is a quantum Wiener process, and its adjoint Rt B † (t) = t0 b† (τ )dτ . They satisfy the commutation relations [4] [B(t), B † (τ )] = min(t, τ ),

[B(t), B(τ )] = 0.

djt (X) = jt (G(X))dt+dB † (t)jt ([X, L])+jt ([L† , X])dB(t). (5) Here G refers to the Lindblad generator [4]

Under a common assumption that the initial state of the field is a vacuum field, the process B(t) is analogous to the standard Wiener process and the process b(t) is analogous to a Gaussian white noise with zero mean. For convenience, we summarize the Ito rules for the quantum infinitesimal increments of B, B † in a vacuum state [11] which will be used in subsequent calculations, dB(t)dB † (t) = dt, dB(t)dB(t) = 0,

2.2

dB † (t)dB(t) = 0, dB † (t)dB † (t) = 0.



G(X) = −i[X, H] + LL (X),

(6)

and the notation LL (X) refers to the Lindblad superoperator defined as LL (X) =

(3)

1 † 1 L [X, L] + [L† , X]L. 2 2

Quantum stochastic differential equations have been widely used in the analysis and control of Markovian quantum systems [4,17,42,22].

Quantum stochastic differential equation

Markovian quantum systems have been extensively investigated since they are suitable models for many physical systems. For example, optical modes trapped in a cavity (see Fig. 1) probed by a white noise field exhibit Markovian dynamics.

2.3

Input-output relations

When a quantum white noise field passes through a quantum object and interacts with it, the resulting field

3

to internal modes of the filter. Similarly, in this paper we assume that the internal modes of the ancillary system correspond to dynamics of the non-Markovian environment. However different from the classical case, we show that the non-Markovian system behaviour and the system backaction on the environment can be explained using a direct coupling mechanism of environment-system interactions, within an augmented Markovian system picture.

is called an output field. It can be observed via measurement, e.g., homodyne detection [11]. Mathematically, the output field is described as Bout (t) = Θ† (t)B(t)Θ(t). The quantum infinitesimal increment for the output field can be written as a quantum stochastic differential equation: dBout (t) = jt (L)dt + dB(t), (7) which shows that the output field not only carries information about the quantum object but is also affected by the input noise. As a result, the output field can be utilized in estimating the dynamics of the quantum object [4,44,46]. 2.4

This section begins with introducing a general framework for constructing such an augmented Markovian system model in Section 3.2. Next in Section 3.3, we show that the class of proposed augmented models is sufficiently rich in a sense that, given an environment with a rational power spectral density of its quantum colored noise, an ancillary linear quantum system model for this environment can be constructed to match that spectrum. This implies that a large class of non-Markovian environments can be modelled (exactly or with a sufficient accuracy) in terms of linear quantum systems consisting of quantum harmonic oscillators.

Master equation

While this paper is exclusively setup in the Heisenberg picture, it is worth reminding about an alternative formulation known as the Schr¨ odinger picture. In the Schr¨ odinger picture, the state of a quantum system can be described by a wave function |ψi which is a complex vector in the Hilbert space h. Based on the wave function |ψi, a trace class operator ρ = |ψihψ| on h is defined known as the density matrix of the quantum system [44].

3.2

In contrast to the Heisenberg picture introduced in the previous subsections, in which the operators evolve in time and the states are time independent, the states of a quantum system evolve in time while the operators are time independent in the Schr¨ odinger picture. Of course, these two alternative representations of the quantum evolution are equivalent [5,44].

3.2.1

Gp = (I, Lp , Hp ),

3.1

(9)

where Hp is the Hamiltonian of the principal system and Lp is the coupling operator vector of the principal system with respect to a probing field defined on a Fock space Fp . The operators Hp and Lp are defined on a system’s Hilbert space hp . Note that the coupling operator Lp does not describe how the system is coupled with the environment, but allows the principal system to be probed for measurement, by shining an input field through probing channels and observing the output of the probing field so as to construct quantum filters.

(8)

where the adjoint of the Lindblad superopertor L∗L is calculated as L∗L (ρ(t)) = 21 [Lρ(t), L† ] + 21 [L, ρ(t)L† ]. 3

The (S, L, H) description of the augmented Markovian system model

Consider a quantum system operating in a nonMarkovian environment. This system, which we call the principal system, has the following (S, L, H) description

The density matrix ρ of a Markovian quantum system characterized by a triple (S, L, H) and interacting with a quantum white noise obeys a so-called master equation ρ(t) ˙ = −i[H, ρ(t)] + L∗L (ρ(t)),

The general augmented Markov system framework

An augmented model for non-Markovian systems

The principal quantum system interacts with its environment by exchanging energy with the environment, i.e., they mutually influence each other. This energy exchange is captured by a direct interaction Hamiltonian

Preliminary remarks

Hpa = i(c† z − z † c),

A non-Markovian quantum system is a quantum system in an environment with memory, and is typically regarded as a quantum system disturbed by a quantum colored noise. In this paper we propose to model such environments as ancillary quantum systems excited by quantum white noise fields. This approach resembles the classical approach to modelling colored noise signals using shaping filters where the shape of the spectrum is related

(10)

where the operator vector c describes the environmental effect, and z is a coupling operator vector z defined on the Hilbert space hp of the principal system. In what follows internal modes of the environment are assumed to be stationary. Hence the quantum correla-

4

tion matrix hc(t+τ )c† (t)i is independent of t; here h·i denotes the quantum expectation defined as h·i = tr[·ρa ], where ρa is the initial density matrix of the environment. Also the power spectral density characteristics of the environment can be defined in a standard fashion, as the Fourier transform of the quantum correlation function.

Non-Markovian Environment White noise

Ancillary system Ga

Probing field

Principal system Gp

Definition 1 The power spectral density of a stationary environment operator vector c is defined as Z

+∞

hc(t + τ )c† (t)ie−st dt.

S(ω) =

(11)

−∞

Note that since the definition of Fourier transform for quantum systems is actually the standard inverse Fourier transform but double sided Laplace transform in equation (11) is standard, we have s = −iω.

Fig. 2. A schematic plot of the general augmented system model for non-Markovian quantum systems.

A schematic diagram of the augmented system is shown in Fig. 2. The augmented system GT is defined on the tensor product Hilbert space hp ⊗ha ⊗Fp ⊗Fa . Since this augmented system only interacts with quantum white noise fields, namely with the ancillary white noise field and the probing field, the overall system is Markovian. However, as we will show in the next section, the principal subsystem is not Markovian due to the interaction with the ancillary system.

It is straightforward to verify using singular commutation relations that when c is the quantum white noise b(t) and the environment is Markovian, then S(ω) = 1. On the other hand, when S(ω) defined by equation (11) is not equal to 1, this corresponds to a colored noise c. To model internal modes of the environment, we introduce an ancillary Markovian quantum system with a Hamiltonian operator Ha and a collection of coupling operators with respect to ancillary quantum white noises, combined into a coupling operator vector La . Using the compact (S, L, H) notation, such an ancillary system is denoted Ga = (I, La , Ha ); (12) as noted previously, the scattering matrix is assumed to be identity. Note that the ancillary system evolves on a Hilbert space ha ⊗ Fa , where the ancillary system and the corresponding white noise are defined on the Hilbert space ha and the Fock space Fa , respectively. Also associated with this system is a vector c of operators of the ancillary system defined on the Hilbert space ha . This vector of operators represents the quantum colored noise whereby the ancillary system interacts with the principal system; see (10). As the ancillary system is driven by a quantum white noise, it can be thought of whitening the quantum colored noise arising from the non-Markovian environment.

3.2.2

By substituting (13) into (4) the Heisenberg picture stochastic differential equation of the evolution operator U (t) for the augmented system (13) can be obtained to be dU (t) =

GT = (I,

Lp

(14)

dX 0 (t) = −i[X 0 (t), H0 (t)]dt + (LLp (t) (X 0 (t)) + LLa (t) (X 0 (t)))dt + ([X 0 (t), c† (t)z(t)] + [z † (t)c(t), X 0 (t)])dt + dBp† (t)[X 0 (t), Lp (t)] + [L†p (t), X 0 (t)]dBp (t)

! , Hp + Ha + Hpa ).

1 − i(Hp + Ha + Hpa )dt − L†p Lp dt 2 1 † † − La La dt + dBp (t)Lp − L†p dBp (t) 2  +dBa† (t)La − L†a dBa (t) U (t). 

Here, dBp and dBa are the quantum infinitesimal increments for the noise processes of the principal and ancillary system, respectively. Then using (14), the evolution of an augmented system operator X 0 can be defined as X 0 (t) = U † (t)X 0 U (t) which satisfies the quantum stochastic differential equation

The general quantum feedback network theory [13] can now be applied to describe interactions between the principal system and the environment. From this theory, the combined system model representing the principal quantum system Gp and the ancillary system model (12) for the non-Markovian environment interacting via the Hamiltonian (10) has the following (S, L, H) parameters, La

Quantum stochastic differential equations for the augmented system

+ dBa† (t)[X 0 (t), La (t)] + [L†a (t), X 0 (t)]dBa (t), (15)

(13)

5

where H0 (t) = U † (t)(Hp + Ha )U (t) and other timevarying operators are obtained in the same way as X 0 (t).

3.2.3

A master equation for the augmented system

Alternatively, dynamics of the augmented system can be described using a master equation. In particular, it is convenient to use master equations when the principal system is a qubit system involving nonlinear dynamics. This will be shown in section 5.

0

Note that any operator of the augmented system X can be written as X 0 = Xp ⊗ Xa , i.e., a tensor product of a principal system operator Xp and an ancillary system operator Xa . Thus, the generator for the augmented system can be expressed as

For the augmented system (13), the master equation (8) takes the form

0

GT = Gp (Xp ) ⊗ Xa + Xp ⊗ Ga (Xa ) − i[X , Hpa ], (16)

ρ(t) ˙ = −i[Hp + Ha , ρ(t)] + L∗La (ρ(t)) + L∗Lp (ρ(t))

where

+[c† z, ρ(t)] + [ρ(t), z † c],

Gp (Xp ) = −i[Xp , Hp ] + LLp (Xp ), Ga (Xa ) = −i[Xa , Ha ] + LLa (Xa )

(17) (18)

(21)

where ρ(t) is the density matrix of the augmented system. Once again we observe that the state evolution of the augmented system is Markovian, since future values of the density matrix ρ(t) only depend on the present density matrix. One can also obtain the density matrix ρp (t) of the principal system as the partial trace with respect to the ancillary system,

are the generators for the principal system and the ancillary system, respectively. In particular, for X 0 = Xp ⊗ I, i.e., a principal system operator, Eq. (15) reduces to

ρp (t) = tra [ρ(t)].

(22)

dXp (t) = −i[Xp (t), Hp (t)]dt + LLp (t) (Xp (t)) 3.3


+ c† (t)[Xp (t), z(t)] + [z † (t), Xp (t)]c(t) dt +dBp† (t)[Xp (t), Lp (t)]

+

[L†p (t), Xp (t)]dBp (t) (19)

Linear quantum systems models for non-Markovian environments with rational power spectral densities

We now demonstrate that the proposed modelling of non-Markovain systems is sufficiently rich in a sense that for a broad class of environment power spectral densities S(ω), an ancillary system can be constructed whose characteristics match S(ω). Specifically, we show that when the environment has a rational power spectral density, the corresponding ancillary system can be realized within the class of linear quantum systems. Clearly, when S(ω) is not rational, but can be approximated by a rational power spectral density, an approximating ancillary linear quantum system model can be constructed. From a practical perspective, such an approximation is often sufficient and leads to meaningful results as will be demonstrated in Section 5.

with Hp (t) = U † (t)Hp U (t). When X 0 = I ⊗ Xa , i.e., an operator of the ancillary system, we have dXa (t) = −i[Xa (t), Ha (t)]dt + LLa (t) (Xa (t)))dt +([Xa (t), c† (t)]z(t) + z † (t)[c(t), Xa (t)])dt +dBa† (t)[Xa (t), La (t)] + [L†a (t), Xa (t)]dBa (t) (20) with Ha (t) = U † (t)Ha U (t). Note that in quantum mechanics it is conventional to write Xp ⊗ I and I ⊗ Xa as Xp and Xa , respectively.

First, let us consider a linear quantum system comprised of n harmonic oscillators interacting with m channels of quantum white noise fields and obtain an expression of the associated power spectral density. Since our aim is to represent a non-Markovian environment in a form of an ancillary system and such environments do not generate energy, we restrict attention to the class of linear quantum systems whose quantum stochastic differential equations only involve annihilation operators.

One can see form (19) that due to the direct coupling term induced by the ancillary system in the second line of Eq. (19), the principal system will not behave as a Markovian quantum system when it is coupled with the ancillary system. Indeed, the ancillary system operator c can be treated as an input into the principal system generated by the ancillary system. As we noted earlier this input represents a colored noise generated by the environment, which explains a non-Markovian nature of the principal systems dynamics. Also, it can be seen from (20) that the evolution of the ancillary system operator Xa depends on the principal system operator z, which shows that the principal system acts back on the ancillary system. This explains the non-Markovian behaviour of the environment.

To obtain an expression for the power spectral density of a linear annihilation only system, we begin with its (S, L, H) description. Specifically, the Hamiltonian of such a linear system has the form Ha = a† Ωa,

6

(23)

where a = [a1 , a2 , · · · , an ]T is a column vector of annihilation operators with an annihilation operator aj as its j-th component, and a† = [a†1 , a†2 , · · · , a†n ] is the corresponding row vector of creation operators. These operators are defined on the common Hilbert space ha and satisfy the singular commutation relations [aj , a†k ] = δjk , [aj , ak ] = 0, j, k = 1, · · · , n.

has the desired power spectral density S(ω). Note that although the operator (28) is expressed in terms of operators of the system (27), unlike the input defined in Eq. (7) it is not an output field of the system available for quantum measurement. Yet it represents a physical quantity whereby the ancillary system can interact with other quantum systems.

(24)

The diagonal and non-diagonal elements of the Hermitian matrix Ω ∈ Cn×n represent the internal angular frequencies and the couplings between the harmonic oscillators, respectively. Also, the system is coupled with a white noise field, and the corresponding coupling operator La of such a linear annihilation only system with respect to the quantum white noise fields ba (t) can be expressed as La = Na a (25) m×n with a matrix Na ∈ C . Also, according to our standing assumption, the identity scattering matrix is assumed.

The proof of Theorem 2 will be given later in this section. Before proving the theorem, it is instructive to compute the power spectral density of the operator (28). Since the system (27) is to represent internal modes of the environment, its dynamics can be assumed to start from a long time ago. Formally, this means that the initial time t0 when the system (27), (28) was at rest is −∞. Hence, c(t) can be expressed as Z

t

Ξa (t − θ)dBa (θ),

(29)

Ha e−Fa t Ga , t ≥ 0; 0, t 0 [19]. Hence, we can find a factorization for the inverse of P , i.e., P −1 = T † T and thus equation (35) can be reexpressed as

−∞

Z

0

∗

Ξa (λ)e−(−s

)λ

†

Fa + Fa† + Ga G†a = 0,

(32)

This equation shows the new realization (37) obtained from (34) by applying the coordinate transformation T satisfies the physical realizability condition of [22]. Also, it follows from [22] that we can obtain expressions for the Hermitian matrix Ω and the coupling operator La as

dλ

(38)

−∞

= Γ(s)Γ∼ (s).

where Γ∼ (s) = Γ† (−s∗ ) is the adjoint of the transfer function Γ(s). The calculation is analogous to the calculation of the power spectral density for an output of a classical linear system driven by a white noise input [19].

Ω=

It follows from (32) that the spectral factorization method can be employed to obtain a linear quantum system representation of the environment with a positive rational spectral density S(ω).

i (Fa − Fa† ), 2

As an example, let us particularize the result of Theorem 2 for the Lorentzian power spectral density of the form 2 γ0 4

, (40) + (ω − ω0 )2 which commonly arises in solid-state systems [48,34].

Proof of Theorem 2 Given a power spectral density S(ω), we will determine the corresponding Hamiltonian (23) and the coupling operator (25) which define the linear quantum system (27), (28). First we observe that according to [35, Theorems 5, 7], S(ω) can be factorized as in (32) with a stable transfer function

γ02 4

Corollary 3 The power spectral density (40) of the Lorentzian noise can be realized by a single mode linear quantum system γ0 √ da0 (t) = −( + iω0 )a0 (t)dt − γ0 dBa (t), √2 γ0 c0 (t) = − a0 (t), 2

n−1

βn−1 s + · · · + β0 . sn + αn−1 sn−1 + · · · + α0

(39)

These quantities define the system (27) as a linear quantum system. 2

S0 (ω) =

Γ(s) =

La = −G†a a.

(33)

8

(41)

with the annihilation operator a0 .

can be monitored via homodyne detection, and the measurement results can be utilized to construct a filter for the system. In this section we show that when the probing field is applied to a principal system directly interacting with an ancillary system, an augmented-systembased quantum filter, i.e., a whitening quantum filter can be derived to process measurements of the output field. Our derivation is based on the following assumptions regarding the physical apparatus.

PROOF. The Lorentzian spectrum (40) can be factorized as in (32) with Γ0 (s) =

γ0 2

s + iω0 +

γ0 2

.

(42)

The order of the denominator of this transfer function is one, i.e., n = 1, which means the ancillary system can be realized by a single mode linear quantum system. Then, the matrices F0 , G0 , H0 and T reduce to scalars. From (34), we obtain F0 = −(

γ0 γ0 + iω0 ), G0 = 1, H0 = . 2 2

Assumption 5 1. The probing field is a quantum white noise field in a vacuum state, which satisfies a nondemolition condition such that the dynamics of the augmented system can be continuously monitored [4]. 2. The monitored channels are assumed to be coupled with the principal system via the operator Lp . 3. The homodyne detector is perfect with 100% detection efficiency.

(43)

Substituting (43) into the Lyapunov equation (36), we have T † T = γ0 , which can be solved to give √ T = − γ0 .

Definition 6 The quantum filtering problem is to determine an estimate of an observable X 0 (t) which is the conditional expectation

(44)

ˆ 0 (t) = πt (X 0 ) = E[X 0 (t)|Y(t)], X

According to (37), the realization of the system (27) with Fa = −(

√ γ0 γ0 √ + iω0 ), Ga = − γ0 , Ha = − , (45) 2 2

i.e, the projection of X 0 (t) on Y(t), the commutative subspace of operators generated by the measurement results Y (τ ), 0 ≤ τ ≤ t.

satisfies the physical realizability condition (38). Hence, equation (41) corresponds to a physically realizable linear quantum system. Indeed, from (39), the Hamiltonian and the coupling operator of this system can be obtained Ha = ω0 a†0 a0 ,

La =

√

γ0 a0

By applying the existing quantum filtering theory, we immediately obtain the whitening filter for the augmented system derived in Section 3.

(46)

Theorem 7 Under the Assumption 5, a quantum filter for the augmented system can be constructed as

where a0 is the annihilation operator of the system. 2

dπt (X 0 ) = πt (GT (X 0 ))dt − (πt (X 0 Lp + L†p X 0 ) − πt (X 0 )

Remark 4 The central frequency of the Lorentzian power spectral density (40) determines the angular frequency ω0 of the ancillary system, and the bandwidth of the power spectral density determines the system damping rate γ0 with respect to the white noise field. 4

×πt (Lp + L†p ))(dY (t) − πt (Lp + L†p )dt). (48) The measurement process Y (t) induces an innovation process W (t) satisfying dW (t) = dY (t) − πt (Lp + Lp † )dt

Application to quantum filtering

and whose increment dW (t) is independent of πτ (X 0 ), τ ∈ [0, t]. 4.1

Whitening quantum filter for non-Markovian quantum systems PROOF. This whitening filter can be derived by applying the standard orthogonal projection approach developed in [4,3] to the augmented system model GT . 2

In quantum physics, to force a quantum system to generate an output field, it must be excited with a probing field. Typically, such a filed is a quantum white noise field, bp (t). The quadrature of the output field, Y (t) = U † (t)(Bp + Bp† )U (t),

Our next result concerns estimating the density matrix of the augmented system.

(47)

9

Definition 8 The conditional density matrix ρˆ(t) of a quantum system is a density matrix which satisfies the equation πt (X 0 ) = tr[ˆ ρ(t)X 0 ]. (49)

As noted previously, the Hamiltonian of a linear annihilation only principal system is quadratic and can be expressed as Hp = d† Λd, (53)

Remark 9 According to the above definition, the conditional density matrix is a density matrix for which the quantum expectation of X 0 in the Schr¨ odinger picture coincides with the orthogonal projection of X 0 (t) onto the commutative subspace Y(t).

where d = [d1 , d2 , · · · , dn0 ]T is a column vector of annihilation operators for the principal system. Also as before, d† = [d†1 , d†2 , · · · , d†n0 ] is the corresponding row vector of creation operators for the principal system. These operators satisfy the singular commutation relations, cf. (24): [dj , d†k ] = δjk , [dj , dk ] = 0, j, k = 1, · · · , n0 .

Theorem 10 The conditional density matrix ρˆ(t) for the augmented system satisfies the stochastic master equation dˆ ρ(t) = GT∗ (ˆ ρ(t))dt + FLp (ˆ ρ(t))dW

(54)

The diagonal and off-diagonal elements of the Hermi0 0 tian matrix Λ ∈ Cn ×n are determined by the angular frequencies of components of the principal system and their couplings, respectively.

(50)

with FLp (ˆ ρ(t)) = Lp ρˆ(t) +

ρˆ(t)L†p

− tr[(Lp +

Also, the coupling operator with respect to m0 channels of the quantum white noise process dBp (t) and the direct coupling operator are specified as

L†p )ˆ ρ(t)]ˆ ρ(t),

(51) where the superoperator GT∗ = i[Ha + Hp + Hpa , ρˆ(t)] + L∗Lp (ˆ ρ(t)) + L∗La (ˆ ρ(t)) is the adjoint of GT .

Lp = Np d, z = Kp d, 0

0

As described in the previous section, the system interacts with a non-Markovian environment. These interactions can be captured using a model where the principal system is directly coupled with a linear ancillary system. Thus, the quantum stochastic differential equation for the augmented system including both the principal and the ancillary systems can be obtained

In practice, when one is interested in estimating the principal non-Markovian system, it is the conditional density matrix of the principal system ρˆp (t) that will be of interest. Formally, it can be obtained as the partial trace of ρˆ(t), by tracing out the ancillary system

"

˙ dd(t) da(t) ˙

(52)

#

" =

Fp

−Kp† Ha

Ha† Kp

Fa " +

Earlier in Section 3.3 we have proposed representing ancillary systems using linear quantum systems. In this P case, equation (52) reduces to ρˆp (t) = hn|ˆ ρ(t)|ni n where the number of bases |ni for the linear quantum system is infinite. This makes obtaining an exact expression for ρˆp (t) from (52) rather difficult. However, approximating the linear ancillary system with a N -level system will have an effect of truncation, and thus it is possible to calculate an approximation to the partial trace (52) [1]. 4.2

0

respectively, where Np ∈ Cm ×n and Kp ∈ Cr×n .

PROOF. The stochastic master equation for the conditional density matrix defined in (49) can be derived from the whitening quantum filter (48) by using the methods in [4,3]. 2

ρˆp (t) = tra [ˆ ρ(t)].

(55)

#"

Gp

d(t)dt

#

a(t)dt #" # 0 dBp (t)

0 Ga

,

(56)

dBa (t)

where Fp = −iΛ − 21 Np† Np and Gp = −Np† . Also, the output field excited by the probing the principal system with the white noise filed process dBp (t) is dBout−p (t) = Hp d(t)dt + dBp (t)

(57)

with Hp = Np . Suppose that the position quadrature of the output,

Whitening quantum filter for linear non-Markovian quantum systems

1 † (t)) dy(t) = √ (dBout−p (t) + dBout−p 2

In this section, we consider the case where the principal system is a linear quantum system whose description involves only annihilation operators. For such quantum systems, we show that the whitening quantum filter is actually a quantum Kalman filter.

is observed via homodyne detection. Also, since the operators in Eqs. (56) and (57) are not self-adjoint and the coefficients may be complex valued, it is convenient to

10

Note that the noise processes w1 (t) and w2 (t) are correlated, and the covariance matrix of [w1 (t), w2 (t)]T is V t, where

transform Eqs. (56) and (57) into a phase space in terms of position and momentum operators as dx(t) = Ax(t)dt + dw1 (t), dy(t) = Cx(t)dt + dw2 (t)

(58) (59)

" V =

with dw1 (t) = Bdw(t), dw2 (t) = Ddw(t), 

A11 A12 A13 A14



A23 A33 A41

A14 = −i(Kp† Ha − KpT Ha† ), † T † = i(Kp Ha − Kp Ha ), A24 = −Kp† Ha − KpT Ha† , = Ha† Kp + HaT Kp† , A32 = i(Ha† Kp − HaT Kp† ), = A44 = Fa + Fa† , A34 = −A43 = i(Fa − Fa† ) = −i(Ha† Kp − HaT Kp† ), A42 = Ha† Kp + HaT Kp† 

" =

BB T BDT

#

DB T DDT

.

(60)

Theorem 11 The whitening quantum filter for the linear augmented system (58) is a steady-state quantum Kalman filter

A12 = −A21 = i(Fp − Fp† )

A13 = −Kp† Ha − KpT Ha† , A31

T V12 V2

#

We are now in the position to present the whitening quantum filter for computing the estimate x ˆ(t) of the vector of dynamical variables x(t) of the linear augmented system model (58).

   1  A21 A22 A23 A24  A=  , 2  A31 A32 A33 A34    A41 A42 A43 A44 A11 = A22 = Fp + Fp† ,

V1 V12

dˆ x(t) = Aˆ x(t)dt + K(dy(t) − C x ˆ(t)dt),

(61)

with the Kalman gain K = (Vˆ C T + V12 )V2−1 ,



B B B B  11 12 13 14   1  B21 B22 B23 B24   B=  , 2  B31 B32 B33 B34   

obtained from the solution of the algebraic Riccati equation (A − V12 V2−1 C)Vˆ + Vˆ (A − V12 V2−1 C)T T −Vˆ C T V2−1 C Vˆ + V1 − V12 V2−1 V12 = 0.

B41 B42 B43 B44

B11 = B22 = Gp + G†p ,

(62)

B12 = −B21 = i(Gp − G†p )

B33 = B44 = Ga + G†a , B34 = −B43 = i(Ga − G†a ) B13 = B14 = B23 + B24 = B31 = B32 = B41 = B42 = 0 i 1h C= Hp + Hp† 0 0 0 , h2 i D= I 0 0 0

This theorem is obtained by applying the existing quantum Kalman filtering results for linear quantum systems [16,41] to the linear augmented Markovian system (58). The obtained whitening quantum filter is driven by the output y(t). Note that the symmetrized covariance matrix Vˆ is independent of the measurement process.

where

4.3 T

x(t) = [qp (t), pp (t), qa (t), pa (t)] , dw(t) = [dvq (t), dvp (t), d¯ vq (t), d¯ vp (t)]T

An illustrative example: A non-Markovian optical system

The system In this example, we consider a single mode non-Markovian bosonic quantum system disturbed by Lorentzian noise, which can be realized by two coupled cavities as shown in Fig. 3. In this figure, the horizontally oriented cavity is the principal system to be estimated. The vertically oriented cavity is the ancillary system converting white noise to Lorentzian noise. The optical modes in the two cavities are directly and strongly coupled by an optical crystal. A probing field is applied to the principal cavity, whose output is observed via homodyne detection. These measurements are used as the data for a quantum filter.

are the quadrature representations of the operators of the principal and ancillary systems and the probing and quantum white noise fields, respectively. The components of x(t) and w(t) are calculated by applying the " # I I coordinate transformation matrix Π = √12 to −iI iI the corresponding components of the vectors of annihilation operators of the principal and ancillary systems: [qp (t), pp (t)]T = Π[dp (t), d†p (t)]T ,

The Hamiltonian operators of the principal and ancillary cavities are Hp = ωp d†p dp and Ha = ωa a†0 a0 , respectively, with angular frequencies ωp and ωa . They are expressed in terms of respective annihilation and creation

[qa (t), pa (t)]T = Π[a(t), a† (t)]T , [dvq (t), dvp (t)]T = Π[dBp (t), dBp† (t)]T , [d¯ vq (t), d¯ vp (t)]T = Π[dBa (t), dBa† (t)]T .

11

White Noise

1

Homodyne Detector

Output Field

Probing Field Beam Splitter

hqp (t)i and hˆ qp (t)i

Optical Crystal

Principal System Ancillary System

Quantum Filter

0.5 0 −0.5 −1 0

20

40 ωp t

60

80

Fig. 4. Unconditional and conditional means of the position components for the principal system with κ = 2GHz, ωp = ωa = 10GHz, γ0 = 0.6GHz, and γ1 = 0.8GHz.

Fig. 3. The illustrative example involving an optical system. The quantity of interest is the quantum state of the principal cavity. The ancillary cavity plays the role of the internal modes of the environment converting quantum white noise to Lorentzian noise and enabling estimation of the state of the principal cavity using the whitening quantum filter.

dp , d†P

hqp (t)i hˆ qp (t)i

the augmented system is Gaussian [27] and thus, the mean of the system operators m(t) = hx(t)i satisfying m(t) ˙ = A0 m(t) can serve as an estimate of the system dynamics; here the quantum expectation is with respect to the initial quantum state ρ0 , h·i = tr[·ρ0 ]. On the other hand, the estimates conditioned on the homodyne detection data are generated by the quantum Kalman filter (61). We now compare these estimates.

a0 , a†0

and satisfying the corresponding operators singular commutation relations (54) and (24). Then, the coupling operators√of the principal√cavity z and L are specified to be z = κdp and Lp = γ1 dp , with constant κ, γ1 . Also, the coupling operator of the √ ancillary system with respect to the white noise is La = γ0 a0 . The cav√ γ ities interact through the operators z and c = − 2 0 a0 ; see (10). With these definitions, the quantum stochastic differential equations for the augmented system can be expressed in the of form (58)

We choose the parameters of the system as ωp = ωa = 10GHz, κ = 2GHz, γ0 = 0.6GHz, and γ1 = 0.8GHz and assume that the initial mean m(0) = [1, 0, 0, 0]T of the unconditional dynamical variables is the same as the initial conditional expectation for the quantum Kalman filter, i.e., x ˆ(0) = m(0) where the first element of x ˆ(0) is hˆ qp (0)i = 1. In Fig. 4, the red line is the trajectory of the mean of the unconditional position operator qp (t) for the principal system. The oscillations of the curve envelopes are caused by the disturbance of the ancillary system, which indicates that energy is exchanged between the principal and the ancillary system showing non-Markovian characteristics. Compared with the unconditional trajectory, the blue line in Fig. 4 shows the average trajectory of the conditional expectation of the position hˆ qp (t)i obtained by averaging over 10000 realizations. It matches the red line very closely. This shows that on average, the whitening quantum filter estimates dynamics of the unconditional variable of the principal system.

with matrices   √ κγ0 − γ21 ωp 0 2   √ κγ0  γ1  −ω 0   √ p −2 2 A =  − κγ0 , γ0   0 − ω a 2 2   √ − κγ0 γ0 0 −ω − a 2 2 √ √ √ √ B = diag[− γ1 , − γ1 , − γ0 , − γ0 ], √ C = diag[ γ1 , 0, 0, 0], D = [1, 0, 0, 0]. The corresponding operators x and w in this example are the quadrature representations of the operators of the principal and ancillary systems and the probing and quantum white noise processes, respectively: x(t) = [qp (t), pp (t), qa (t), pa (t)]T dw(t) = [dvq (t), dvp (t), d¯ vq (t), d¯ vp (t)]T . Note that y(t) is the position quadrature of the output of the probing field process.

The spectrum with respect to the white noise field Generally, the output field spectrum is indicative of properties of the system. To calculate the power spectral density of the output of the probing field, we recall the quantum stochastic differential equation of the aug-

Whitening filter estimates vs the mean of the principal system Assume the initial state ρ0 of

12

mented system "

ddp (t)

"

# =

da0 (t)

−iωp − √

−

√

γ1 2

κγ0 2

κγ0 2

#"

dp (t)

0.1GHz, γ0 = 0.1GHz, and γ1 = 0.8GHz. When there is no detuning, the noise is strong at the system frequency, as the blue line shows. As the detuning is increased via decreasing the angular frequency of the ancillary system, the spectrum |G2 (˜ ω )|2 is driven away from the system frequency, and its amplitude is reduced as well. This illustrates that the non-Markovian effect generated by the ancillary system becomes weaker as the detuning is increased. When the detuning is large enough, the dynamics of the ancillary system become negligible. This is consistent with the results in [7].

#

dt −iωa − γ20 a0 (t) "√ #" # γ1 0 dBp (t) − , (63) √ 0 γ0 dBa (t)

where dBp (t) and dBa (t) are the probing field and ancillary white noise field processes, respectively. The output equation with respect to the probing field dBp (t) is dBout (t) =

√

γ1 dp (t)dt + dBp (t).

Fig. 5(c) shows the power spectral density |G2 (˜ ω )|2 varying with the damping rate γ0 . As predicted, the bandwidth of the Lorentzian spectrum is broader as γ0 increases.

(64)

By detecting the position quadrature of the output field † dyq (t) = √12 [dBout (t) + dBout (t)], the power spectral density of the output field can be calculated to be S(˜ ω) =

1 (|G1 (i˜ ω )|2 + |G2 (i˜ ω )|2 ) 2

4.4

A qubit is a basic unit of quantum information, and is defined on a two-dimensional complex Hilbert space hq spanned by the Pauli matrices [26]

(65)

where ω ˜ = ωp − ω, and G1 and G2 are the transfer functions from the probing field process and the quantum white noise process to the output field process, respectively. The square of their norms are expressed as |G1 (i˜ ω )|2 = |G2 (i˜ ω )|2 =

Υ(˜ ω) + Υ(˜ ω) + Υ(˜ ω) + 2

1 16 (κ − 1 16 (κ + κγ1 γ02 4 1 16 (κ +

" σx =

γ1 )2 γ02 , γ1 )2 γ02

0 1 1 0

#

" ,

σy =

0 −i

#

" ,

i 0

σz =

1 0 0 −1

# . (67)

In addition, the ladder operators for the qubit system "

(66)

γ1 )2 γ02

with Υ(˜ ω ) = ( γ41 + ω ˜ 2 )(˜ ω − ∆)2 + ∆ = ωp − ωa .

Quantum filter for a non-Markovian single qubit system

σ− = γ02 ω ˜2 4

− κγ20 ω˜ (˜ ω − ∆),

0 0 1 0

#

" , σ+ =

0 1

# (68)

0 0

are utilized to describe a state flip between the ground state |0i and the excited state |1i. The ladder operators can also be used to describe the interaction with external systems, e.g., in the Jaynes-Cummings model [36].

Although the total output spectrum given in (65) is flat due to the passivity properties of the system [45], we can apply a coherent probing field whose strength is much higher than the strength of the ancillary quantum white noise and thus the spectrum |G1 (˜ ω )|2 can be observed. This allows us to calculate the spectrum |G2 (˜ ω )|2 which reflects the influence of the ancillary system on the output field.

The Hamiltonian of the single qubit system we consider is given as ωq Hq = σz , (69) 2 where ωq is the qubit working frequency characterizing the energy difference between the ground state and and the excited state.

Fig. 5(a) shows the power spectral density |G2 (˜ ω )|2 varying with the coupling strength κ. Here, we assume that there is no detuning (i.e., ∆ = 0) and the damping rates of the principal system to the probing field and of the ancillary system to the quantum white noise field are γ1 = 0.8GHz and γ0 = 0.1GHz. As the coupling strength κ is increased, the amplitude of the noise spectrum is increased, which means that the disturbance for the principal system becomes stronger.

A non-Markovian qubit system can also be represented as an augmented system (13), where the Hamiltonian of the principal system is replaced by Hq and the coupling operator Lp and the direct coupling operator z are defined on the Hilbert space hq . Thus, we can write down a quantum filter for the qubit system in the Heisenberg picture as Eq. (48). However, these filter equations are infinite dimensional. Hence, for a non-Markovian qubit system, it is useful to calculate the evolution of the unconditional and conditional states in the Schr¨ odinger

The power spectral density |G2 (˜ ω )|2 varying with the detuning ∆ is plotted in Fig. 5(b) with parameters κ =

13

1

κ = 0.1 κ = 0.3 κ = 0.5 κ = 0.7

0.8

0.4

0.4

0

0.2

0.1

0.2

−0.4

−0.2

0 ω ˜

0.2

0.4

0 −1

0.6

2

γ0 γ0 γ0 γ0

0.3 |G2 (˜ ω )|2

|G2 (˜ ω )|2

|G2 (˜ ω )|2

0.3 0.6

0.4

∆=0 ∆ = 0.5 ∆=1 ∆ = 1.5

= 0.1 = 0.6 = 1.6 = 3.6

0.2

0.1

−0.5

0

0.5 ω ˜

1

1.5

0 −4

2

−3

2

−2

−1

0 ω ˜

1

2

3

4

2

(a) |G2 (˜ ω )| varying with κ where ∆ = (b) |G2 (˜ ω )| varying with ∆ where κ = (c) |G2 (˜ ω )| varying with γ0 where ∆ = 0, γ0 = 0.1GHz, and γ1 = 0.8GHz. 0.1GHz, γ0 = 0.1GHz, and γ1 = 0.8GHz. 0, κ = 0.1GHz, and γ1 = 0.8GHz. Fig. 5. The variation of the spectrum |G2 (˜ ω )|2 versus varying parameters. White Noise

Homodyne Detector

0 .0 5 0 .0 0 − 0 .0 5




ˆ z (t) σ




σz (t)

− 0 .1 5

Probing Field Beam Splitter




− 0 .3 5 0 1 .0 0 .8

Qubit System Ancillary System

Quantum Filter

20

40

60

ˆ x (t) σ




0 .0




σx (t) m

σx

20

40

60

100


ˆ y (t) σ

0 .4




0 .0




− 0 .8 0

(t)

80

σy (t) m

σy

− 0 .4

picture using the master equation (21) and the stochastic master equation (50), respectively.

100


− 0 .4

Fig. 6. Illustrative example of a non-Markovian qubit system disturbed by Lorentzian noise.

(t)

80

0 .4

− 0 .8 0 1 .0 0 .8

m

σz

− 0 .2 5

20

40

60

80

(t)

100

ωq t

As an example, consider a single qubit system disturbed by Lorentzian noise arising from an ancillary system as shown in Fig. 6. As in the previous section, the ancillary system is a single mode linear quantum system with the √ Hamiltonian ωa a†0 a0 , the coupling operator La = γ0 a0 , √ γ and the fictitious output c = − 2 0 a0 . Here, the damping rate of the ancillary system with respect to the quantum white noise field is γ0 = 0.6GHz. In addition, the qubit is coupled with the ancillary system √ and the probing field through the direct operator z = κ1 σy and the coupling √ operator Lq = γq σx , respectively, with the direct coupling strength κ1 = 1GHz and the damping rate of the qubit γq = 0.8GHz., The single qubit system is initialized in a state ρq (0) = 12 (I + σx ), where ρq is the density matrix of the qubit and I is the 2 × 2 identity matrix. The angular frequency of the ancillary system is equal to that of the qubit, ω0 = ωq = 10GHz. Note that the dynamics of the ancillary systems cannot be eliminated via the adiabatic elimination which is only valid for the off-resonant case, i.e., when there exists a large detuning between the qubit system and the ancillary systems [7].

Fig. 7. The observables for the qubit system in both non– Markovian and Markovian cases. The unconditional and averaged conditional expectation of the observables in the non-Markovian case are denoted as hσ. (t)i (green lines) and hˆ σ. (t)i (blue lines), respectively. For the Markovian qubit, the unconditional expectations of the observables are denoted as hσ.m (t)i (red lines).

conditional expectation of the observables for the qubit system can be calculated as hˆ σ i = tr[(σ ⊗ I)ˆ ρt ], where σ ∈ {σx , σy , σz }. Here, I is the identity matrix defined on the Hilbert space of the ancillary system. The averages of the three conditional expectations hˆ σx i, hˆ σy i and hˆ σz i are plotted as blue lines; they are obtained by averaging over 500 realizations of the trajectories. The green lines represent the unconditinal expectations hσx i, hσy i and hσz i which are obtained from the master equation for the augmented system (21). Once again, we observe that the whitening quantum filter can estimate the nonMarkovian evolution of the single qubit system. To compare with the non-Markovian trajectories, the unconditional expectation values of the observables σx , σy , and σz for the qubit system in the Markovian case are also plotted as the red lines in Fig 7, where the qubit is directly open to the quantum white noise field and the probing field. In this case, the system dynamics obey a

Fig. 7 shows the evolution of both the unconditional and averaged conditional expectation values of the observables σx , σy , and σz for the non-Markovian qubit system. The conditional state ρˆ(t) for the augmented system can be obtained from the quantum filter (50) and thus the

14

White noise White noise

Ancilla 1

Quantum dot

Cavity

Ancilla 2

Quantum dot

Measurement

Original system

Cavity

Measurement

Augmented system

Fig. 9. Block diagram of the hybrid solid-state system considered in [10].

augmented system model to explore the effect of the colored noise assumption. A block diagram of the hybrid solid-state system in [10] is shown in Fig. 9. In the Markovian system model considered in [10], a double quantum dot qubit is directly coupled with a superconducting waveguide resonator (i.e., a cavity) which can be described by a Jaynes-Cumming Hamiltonian as

Fig. 8. (a) Resonator frequency shift ∆ν0 and (b) broadened resonator linewidth κ∗ for the hybrid solid-state system, where the experimental data and the curves for the Markovian system model are obtained from [10] and the non-Markovian curves are calculated using an augmented Markovian system model.

Hd = (ν0 − νd )a†r ar +  + (a†r + ar ) 2

Markovian master equation

νqb − νd σz + gc sinθ(a†r σ− + σ+ ar ) 2 (70)

where the angular frequency of the resonator is ν0 = p 6.755GHz and the frequency of the qubit is νqb = 4µ2 + δ 2 with µ = 4.5GHz. Here, δ is the detuning frequency between the qubit states, which can be varied when measuring the resonator. The coupling strength gc = 0.05 × 2πGHz between the resonator and the qubit can be modulated via a sine function sinθ = √ 2µ2 2 . The annihilation and creation opera-

ρ˙ q (t) = −i[Hq , ρq (t)] + L∗z (ρq (t)) + L∗Lq (ρq (t)). The comparison shows that not only does the qubit in the Markovian case damp faster than that in the nonMarkovian case but also the stationary states of the qubit in the two cases are different.

4µ +δ

5

Application to an experiment on a hybrid solid-state quantum device

tors for the resonator are denoted as ar and a†r , respectively. The symbols νd and  represent the frequency and amplitude of the driving field, which is a weak and fixed strength field; see [9] for more details.

In some experiments involving solid-state systems, Markovian system models cannot completely explain some subtle phenomena. For example, in the experiment for effectively measuring a double quantum dot qubit in a superconducting waveguide resonator [10], the experimental data on the broadened resonator linewidth under suitable parameters disagrees with a calculation based on a Markovian system model. This discrepancy in [10] was predicted to be caused by colored noise. In this section, we assume that the discrepancies in the experiment are caused by colored noise and apply our

Furthermore, the qubit is coupled with one dissipative channel and one dephasing channel which q are character√ γ ¯z ized by coupling operators γ¯− σ− and 2 σz where 0 0 γ¯− = sin2 θ¯ γ− + cos2 θ¯ γz0 and γ¯z = cos2 θ¯ γ− + sin2 θ¯ γz0 0 0 with γ¯− = 3.3 × 2πGHz and γ¯z = 100 × 2πMHz. In addition, the cavity is probed √ by a field; the correspond¯ ar with κ ¯ = 2.6 × 2πMHz. ing coupling operator is κ Hence, the dynamics of this hybrid system obey the mas-

15

ter equation ρ¯˙ (t) = −i[Hd , ρ¯(t)] + L∗√κ¯ ar (¯ ρ(t)) ρ(t)) + L∗√ γ¯z + L∗√γ¯− σ− (¯ 2

σz

(¯ ρ(t)),

(71)

where ρ¯(t) is the density matrix for the resonator and qubit system. With this model, the transmission amplitude through the cavity is calculated as har i = tr[ar ρ¯s ] where ρ¯s represents the steady state of Eq. (71). The shift of the resonance frequency ∆ν0 and the broadened linewidth of the cavity κ∗ can be obtained from the square of the transmission amplitude, which are plotted as red lines in Fig. 8 (a) and (b), respectively. Compared to the experimental data plotted as blue dots, the shift of the resonance frequency curve in Fig. 8 (a) is in good agreement but the broadened linewidth curve in Fig. 8 (b) has a large discrepancy. This discrepancy is potentially caused by colored noise as conjectured in [10].

Fig. 10. Colored noise spectrum generated by the two ancillary systems.

written as √ H0d = Hd +

X

(i

j=1,2

κ ¯ j γ¯j (σ+ aj − a†j σ− ) + (νj − νd )a†j aj ), 2

(72) where a1 , a2 and a†1 , a†2 denote the annihilation and creation operators for the ancillary systems, and the respective angular frequencies of the two ancillary systems are ν1 = 6.755GHz and ν2 = 1.2GHz. The coupling strengths between the ancillary systems and the quantum dot are κ ¯ 1 = 0.5¯ γ− and κ ¯ 2 = 1.125¯ γ− , respectively. The coupling operators of the ancillary systems √ with√respect to quantum white noise fields are γ¯1 a1 and γ¯2 a2 with γ¯1 = 35MHz and γ¯2 = 20GHz, respectively. Thus, the dynamics of the augmented system can also described by a master equation

To explore the reason for the discrepancy in the broadened linewidth curves and obtain better curve matching, an augmented system model is utilized to describe the system dynamics. We assume that the dissipative channel of the quantum dot in the original system is a colored noise channel. However, since the spectrum of the colored noise is unknown for this hybrid solidstate system, it is difficult to realize the ancillary system in the augmented system by using the spectral factorization method. From Corollary 3, we have known that Lorentzian noise can be generated using a singlemode linear quantum system. When a suitable number of single-mode linear ancillary systems are coupled with the quantum dot through an identical operator of the quantum dot, the Lorentzian noises generated by the ancillary systems can be combined to approximate an arbitrary colored noise. We have tried several such approximations; the results of these trials are given in the appendix.

ρ0 (t)) ρ¯˙ 0 (t) = −i[H0d , ρ¯0 (t)] + L∗√κ¯ ar (¯ X + ρ0 (t)) + L∗√ γ¯z L∗√γ¯j aj (¯ j=1,2

2

σz

(¯ ρ0 (t)),

(73)

where ρ¯0 (t) is the density matrix of the augmented system. The transmission amplitude through the cavity is now har i0 = tr[ar ρ¯0s ], where ρ¯0s is the steady state of Eq. (73), and thus with the modified model the shift of the resonance frequency and the broadened linewidth of the cavity can be obtained as well.

One case we considered is where only one ancillary system is coupled to the quantum dot. When the ancillary system is resonant or off-resonant to the cavity, the peak values of the broadened resonator linewidth κ∗ can be modified to provide a good fit to the experimental data. However, the peak values of the resonator frequency shift ∆ν0 vary in an opposite direction.

After several trials using the two-ancillary model, we obtained a broadened linewidth which matches the experimental data significantly closer then the Markovian model; see the green lines in Fig. 8. At the same time, the curve for the shift of the resonance frequency deviates only slightly from the experimental data. The spectrum of the colored noise generated by the two ancillary systems is shown in Fig. 10, it has a double Lorentzian shape. One Lorentzian spectrum is sharp with an identical frequency to the cavity and the other one is very broad with a center frequency far away from that of

Next, we considered the situation where the quantum dot is coupled to two ancillary systems, one of them was a resonant system and another one was an off-resonant ancillary system. The block diagram for the model with two ancillary systems is shown in Fig. 9. This modified model with refined parameters can be described as follows. The Hamiltonian of this modified model can be

16

both the resonator and the quantum dot. These results indicate that the discrepancies in [10] can indeed be attributed to colored noise. Possibly even better results could be achieved if more ancillary systems were utilized. However, the computation time increases considerably in this case due to the dimensions of the augmented system. 6

[7] A. C. Doherty and K. Jacobs. Feedback control of quantum systems using continuous state estimation. Phys. Rev. A, 60:2700–2711, Oct 1999. [8] D. Y. Dong and I. R. Petersen. Sliding mode control of twolevel quantum systems. Automatica, 48(5):725 – 735, 2012. [9] T. Frey. Interaction between quantum dots and superconducting microwave resonators. PhD thesis, ETH Z¨ urich, 2013. [10] T. Frey, P. J. Leek, M. Beck, A. Blais, T. Ihn, K. Ensslin, and A. Wallraff. Dipole coupling of a double quantum dot to a microwave resonator. Phys. Rev. Lett., 108:046807, Jan 2012.

Conclusions

In this paper, we have presented an augmented Markovian system model for non-Markovian quantum systems. Also, a spectral factorization approach has been used to systematically represent a non-Markovian environment by means of linear ancillary quantum systems. Importantly, direct interactions between the ancillary and principal systems are introduced, which result in nonMarkovian dynamics of the principal system. Next, we demonstrated that using this augmented system model, a whitening quantum filter can be constructed for continuously estimating dynamics of non-Markovian quantum systems. Such a filter has been derived for both linear and qubit principal systems. The proposed augmented Markovian system model has also been applied to explore the structure of unknown colored noises in the experiment for the hybrid resonator and quantum dot system.

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The augmented Markovian system model is defined on an extended Hilbert space whose dimension is determined by the number of the linear ancillary systems. When the dimension of the extended Hilbert space is large, the calculation speed for the whitening quantum filter may be slow. Hence, for future studies, it is worthwhile to explore more efficient techniques for representing colored noise by means of ancillary systems which can improve the computational speed of the whitening quantum filter.

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Additional simulation results for the hybrid solid-state system

We also consider three possible cases which might happen to the hybrid solid-state system in section 5. The first case is that the quantum dot is distrubed by Lorentzian noise where it directly interacts with one linear ancillary system with a resonant frequency, i.e., ν1 = ν0 . The results for this case are plotted in Fig. A.1, where the experimental data and the curves for the Markovian case are plotted as blue dots and red lines, respectively. When the damping rate of the ancillary system is γ¯1 = 10MHz, the curve for the broadened resonator linewidth κ∗ in Fig. A.1(b) can get closer to the experimental data. However, the peak value of the corresponding curve for the resonator frequency shift ∆ν0 in Fig. A.1(a) is decreased too much compared with that of the experimental data. As increasing the damping rate of the ancillary system, e.g., γ¯1 = 100MHz, the curves approach to that of the Markovian case.

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In addition, the case that the quantum dot is coupled to both one ancillary system and a Markovian dissipative channel is considered. The frequency of the ancillary system is ν1 = ν0 and the damping rates of the ancillary system is γ¯1 = 35MHz. The damping rate of the quantum dot with respect to the Markovian dissipative channel is kept the same as γ¯− . In this case, both the peak values of the resonator frequency shift ∆ν0 and the broadened resonator linewidth κ∗ are increased.

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In the first two cases, the peak values of the broadened resonator linewidth κ∗ can be modified to approach to the experimental data . However, the peak values of the resonator frequency shift ∆ν0 vary in an opposite direction. Hence, we consider to couple the quantum dot to one resonant and one off-resonant ancillary systems and thus obtain the results in section 5.

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Fig. A.1. (a) Resonator frequency shift ∆ν0 and (b) broadened resonator linewidth κ∗ for the hybrid solid-state system, where the quantum dot is coupled with one ancillary system with a resonant frequency, i.e., ν1 = ν0 .

Fig. A.2. (a) Resonator frequency shift ∆ν0 and (b) broadened resonator linewidth κ∗ for the hybrid solid-state system, where the quantum dot is coupled with one ancillary system with an off-resonant frequency, i.e., ν2 = 1.2GHz.

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Fig. A.3. (a) Resonator frequency shift ∆ν0 and (b) broadened resonator linewidth κ∗ for the hybrid solid-state system, where the quantum dot is coupled to both one ancillary system and a Markovian dissipative channel.

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